Harry's main presence in the Spiderverse has been as a friend to Peter Parker, as well as the rest of that social network (e.g. Gwen, Mary Jane, Flash). He was often burdened by emotional vulnerabilities that usually landed him in some hot spots. Let's start with Amazing Spider-Man #97 where a distraught Harry begins popping pills to cope with his break-up with Mary Jane Watson.
His drug use lands Harry in the hospital and Peter uses this against Norman (who had resumed the role of the Green Goblin) as a means of emotionally jarring Norman enough to cease embracing his Green Goblin persona.
As seen in Amazing Spider-Man #98, Peter's tactic against Norman worked.
Harry's battles with drug-usage continued in Amazing Spider-Man #121, unfortunately this incident has the opposite affect Norman. The collapse of Harry's health coupled to an economic down-turn for Osborn Industries causes Norman to relapse into the Green Goblin persona. This eventually leads to Gwen Stacy's death at the Brooklyn Bridge.
Somehow Harry managed to drag his drug-addled body from bed and witness his father's death (from his own glider) during a battle with Spider-Man in Amazing Spider-Man #122. After Spidey left the scene, Harry removed the Green Goblin attire from his father which effectively exonerated him from any wrongdoing as the villainous Goblin. This marks the official 'first time' when Harry was aware that his father was the Green Goblin.
Harry's effort worked, the authorities (for the most part) were convinced Norman died an innocent man during a confrontation between the Green Goblin and Spider-Man (as seen in Amazing Spider-Man #123).
Harry then returned to the Goblin Hideout and began his preparation to taken on the mantle of the Green Goblin and avenge his father's death; Norman watched this play out and used the opportunity to hide in exile in Europe (as seen in Spider-Man: The Osborn Journal).
Harry then went off into seclusion for a bit, as seen in Amazing Spider-Man #126, and his confused and vulnerable emotional state brought forth his assumption of the role of the Green Goblin.
In Amazing Spider-Man #137, Harry unveils his plot against Peter; showing the kidnapped victims Flash, MJ, and Aunt May...each with a bomb suspended above their heads.
Eventually Peter defeats Harry and the young Osborn was taken in by the authorities. His claims that Peter Parker is Spider-Man are largely ignored due to his psychotic breakdown. Peter protected Harry's secret as the Green Goblin from the authorities as well (by removing his Goblin attire).
As a result of his psychosis Harry begins to see a psychiatrist named Bart Hamilton. Unfortunately for Harry he does not receive much beneficial treatment as Bart uses hypnotic regression to obtain a detailed history of the Green Goblin (as explained in a flashback in Amazing Spider-Man #180).
Harry was taken hostage by Hamilton as the psychiatrist went on a Green Goblin crime-spree.
Harry refutes the Goblin mantle and Hamilton's own lunacy causes a bomb to go-awry and lead to his death.
Harry takes a fall in the course of the ruckus, causing him to suffer yet another amnesia spell similar to what used to happen to Norman. Harry forgets about both his and his father's Green Goblin past.
Ignorance was bliss for Harry, however that did not remain the case. In Amazing Spider-Man #238, Roderick Kingsely stumbles upon one of Norman's hideouts...uncovering a treasure trove of Goblin secrets and weapons.
Harry becomes privy to all of Roderick's break-ins at Osborn Industries properties...he immediately assumes the worst (as seen in Amazing Spider-Man #239). It's interesting here that Harry was unaware that his father was the Green Goblin (due to the amnesia), yet he still recalled the secretive nature with which his father operated.
Roderick begins his blackmail of Harry in Amazing Spider-Man #249, as the Hobgoblin uses Norman Osborn's secret journals to extort his old cronies (J. Jonah Jameson being amongst those cronies).
Harry confides in Peter Parker regarding his father's dark past. This marks this second time Harry became aware that his father was the Green Goblin.
Harry also acknowledged that his father was responsible for the death of Gwen Stacy.
Roderick became obsessed with surpassing Norman's run as the Green Goblin and the key to doing so was the access he had to all of Norman's journals (as seen in Amazing Spider-Man #250).
Unfortunately for Roderick those journals go up in flames after a battle with Spider-Man (as seen in Amazing Spider-Man #251).
Despite Peter's reassurance that Norman's secret was safe (because of the destruction of the journals), Harry continued to struggle with the demons of his father's past.
In Amazing Spider-Man #261 Harry desperately searched through his father's belongings and eventually found a journal in Norman's old roll top desk. The journal contained the locations of all Norman's Goblin Hideouts.
The Hobgoblin took the journal from Harry...then took him prisoner.
Once in the Hobgoblin's custody, Harry's memories of the Green Goblin returned...prompted by the sight of the Hobgoblin's armory.
Harry attempted to use the Hobgoblin's weapons against the villain...he was not very successful.
Thankfully Spidey saved the day and bailed Harry out of trouble; M.J. and Liz were captive too, Spidey saved them as well.
The lost Osborn journal ended up proving useless to the Hobgoblin anyway who had already raided all the hideouts listed therein.
Norman became a grandfather when Liz and Harry welcomed Norman "Normie" Osborn to world in Amazing Spider-Man #263.
We know from Peter Parker Spider-Man (Vol. 2) #25 that Norman briefly returned from his European exile to witness the birth of his grandson. He was able to come and go without being seen by MJ, Peter or his family.
The Hobgoblin was not through with Harry though and in Spectacular Spider-Man #130 he returned to cause more turmoil for Norman's son. In a relatively convoluted plan, the Hobgoblin decided to poison Harry as a means of goading Spider-Man into doing his dirty work.
Harry then contacted Peter (at this point unaware that Peter is Spider-Man) so that he could get Spidey to the Washington West Warehouse.
No need to bore with all the details from this story, but suffice to say Spidey was able to obtain the antidote and save Harry from being killed by the Hobgoblin's poison.
The curse of the Green Goblin was always too much for Harry to escape and although he had been spending his time as a devoted family man, the memories of the Green Goblin slowly began to return in Spectacular Spider-Man #146. Harry had returned from New Jersey to his old home in Long Island as kept abreast of Osborn Industries. He was troubled by a dream involving himself, his father and a grinning monster. By the end of that story, he was even seeing the Goblin whilst awake. Although he was still not quite certain who the Green Goblin was (he should be though, b/c he had flashbacks of the Green Goblin back in Amazing Spider-Man #261).
As if Harry needed more provocation, in Web of Spider-Man #47 the Hobgoblin (this time Jason Macendale) attacked Harry and his family in search of the Goblin Formula. Macendale was most likely aware of Harry's connection to the original Green Goblin when he came into possession of the all the original Hobgoblin's weapons/information. Macendale's history is covered briefly here.
The trauma brought on by the Hobgoblin's attack led to a flood of memories returning to Harry. He recalled from his father's old notes (read long ago) that there was a stash of Green Goblin gear hidden under the southwest window of the Osborn household in Hicksville, Long Island (where Harry had returned to). He once again became the Green Goblin, this time to defend himself and his family...the same way Norman would have done (according to Harry).
In Amazing Spider-Man #312, Harry, along with some help from Spider-Man, defeated the Macendale-Hobgoblin. He told the Hobgoblin that the formula was gone (destroyed years ago) and frightened him so that he never attempted to harm Liz or Normie again.
Despite being through with the Macendale-Hobgoblin, Harry kept the Goblin close to him. Probably a little too close as M.J. (technically Normie) stumbled upon a secret Goblin cache in Web of Spider-Man #65.
Spidey then found himself fighting alongside the Goblin (and Molten Man) against Tombstone and his men in Web of Spider-Man #66.
At first Spidey thought the worst of Harry's mental state.
In Web of Spider-Man #67 Harry explained his rationale for re-becoming the Green Goblin.
Of course things were a bit more complicated than simply Harry wanting to be a hero. He also wanted to vindicate his father's name and the name of the Green Goblin. The interesting part here was that Norman hadn't yet been officially outted as the Green Goblin at that point. A public campaign to redeem Norman and/or the Green Goblin may have been overkill.
Peter disagrees with Harry's plan to salvage Norman's past.
Interestingly here Peter mentions that Norman spent years training to become the Green Goblin.
For those that think Norman had 'good side' that was supressed by the Goblin Formula, here Spidey tells Harry "there was nothing good about Norman Osborn."
During the course of their fight, Harry smashes into a boat...and begins to feel a little outmatched by Spidey.
Peter convinces Harry to leave the heroics to the super-powered individuals.
Peter uses Liz and Normie to convince Harry not risk his life as a heroic Green Goblin. Interestingly, Harry refers to Spider-Man as Peter in the last panel...although Spidey doesn't hear him say that.
Harry struggled to keep a lid on his demons as the Green Goblin and in Spectacular Spider-Man #179 the ghost of Norman Osborn returned (literallly!) to wreak havoc on Harry's life.
Norman laid the guilt on pretty thick (technically it was all inside Harry's head...we think), which prompted Harry to resume the mantle of the Green Goblin.
Harry then attacked Peter for real...no ghosts this time.
Peter refused to fight Harry, but Norman encouraged him to attack (as seen in Spectacular Spider-Man #183).
Harry did not listen to his father and he refused to harm Peter.
Harry's next two appearances as the Green Goblin were a little quirky. The first was featured in Amazing Spider-Man: Deadball; and Harry's relapse was initiated by a very unlikely coincidence. That coincidence: He witnessed Peter catch a foul ball on TV whilst holding Normie in his arms.
So Harry does the logical thing and dons the Goblin attire and flies his glider to Montreal to attack the Expos and Peter Parker!
Peter changes into his Spider-Man gear and defeats Harry.
Harry evades the authorities by simply flying off into the sky; Peter could not follow since he was out of webbing.
The second of the two quirky Goblin appearances was featured in Amazing Spider-Man: Pro-Action #1. This time Harry showed up on the attack against Peter seemingly unprovoked. His mission was to kill Peter and everyone close to him for what Spider-Man did to his father. He dosed Spidey with a gas that nullified his Spider-Sense; most likely this was the same gas Norman used on Peter back when he unmasked him.
Peter turned the tables on Harry though and used a trick that he successfully pulled on Norman back during Harry's first overdose; He brought Harry to Normie. The sight of his son caused Harry to lose control.
Their fight ended with Harry seemingly dying as he and Goblin Glider exploded in a fiery blast.
Not long after that, Harry officially adopted the Goblin mantle when he dosed himself with the Goblin Formula in Spectacular Spider-Man #188.
With his descent into madness nearly complete, Harry invited Peter to his home to gloat about how great things were in his life (as compared to Peter's). Harry had gone over the deep end (as seen in Spectacular Spider-Man #189) and he planned to bring Peter right along with him. Harry (technically) had kidnapped his family members in order for them to be alongside him at the table.
And Harry ended up in the Vault. This was one of those rare instances when an Osborn was publicly arrested as the Green Goblin.
But Harry did not stay in custody long, he returned, alongside Liz and Normie, to Peter's life in Spectacular Spider-Man #199.
Harry's lawyers (Brodsky and Youdelman) got him off the hook, apparently the authorities were also trying to tie Norman's crimes onto his rap-sheet as well. At this point, Norman was still only 'allegedly' the original Green Goblin.
Harry's seemingly normal behavior was obviously a facade as he began sweating profusively as he reminisced over his history with Peter and M.J.
At the conclusion of Spectacular Spider-Man #199 and the beginning of Spectacular Spider-Man #200, M.J. became the Green Goblin's next hostage and he took to that all-too-familiar Brooklyn Bridge. But he didn't bring her up there to kill her, in fact it was just the opposite. He told her she had nothing to fear from the Green Goblin, but two things became abundantly clear in their conversation. For one, Harry harbored a great deal of resentment toward Peter. Two, any mention of Harry's father prompted an immediate emotionally charged response from Harry...he was losing his mind.
He then confronted J. Jonah Jameson and requested that he inform his readership about the gala Harry was throwing to celebrate the opening of The Osborn Foundation. Jameson complied.
Harry clearly wanted (once again) to vindicate his father's (potentially) sullied reputation.
But this time, thanks to another chemical concoction, Harry ends up with the upper hand. And his plan was for both he and Peter to die together in order to rid the world of their collective turmoil. The explosion was set to take place at the Osborn Foundation.
Harry then collapsed on the sidewalk and 'died' whilst Peter, Normie and M.J. watched helplessly.
Harry 'died' having made peace with his best friend Peter Parker.
There were (at least) four posthumous attacks Harry set forth onto Peter.
First, Harry turned out to be the mastermind behind the robotic-constructs that were Peter's long lost parents, Richard and Mary Parker. The panels below are from Amazing Spider-Man #390 and they provide a good synopsis of that particular plot.
Second on the list, Harry had hidden a device in an abandoned Osborn Industries warehouse that Peter inadvertantly stumbled upon in the Spectacular Spider-Man Annual #14. Harry had pre-recorded his memories of the Osborn Family Tree (ie, an Osborn family history). Technically this device wasn't meant strictly for Peter, it was more of 'To Whom it May Concern' and Spidey just happened to be the one to come across it. So it is not a direct shot from Harry at Peter. Interestingly, he claimed to have been somewhat responsible for the explosion that caused his father to become the original Green Goblin. The ramifications of this story are discussed here and here.
The third incident was featured in Spectacular Spider-Man #204. Whilst issuing his last will and testament (in full Green Goblin attire), Harry leaves his mother's jewelry box for Peter as a 'memento of [their] friendship'. Peter, obviously suspicious, doesn't know what to think about the gift. Eventually, guided by his spider-sense, Peter opens the box and finds nothing but a classic Harry "Gotcha!"
Lastly, Harry had engineered the kidnapping of his son Normie with the hope of exposing the youngest Osborn to the Goblin Formula. Harry utilized robotic female Goblinettes to do the dirty work, as seen in Spider-Man: Legacy of Evil. The events of that one-shot are recapped here.
And like his father before him, Harry found a way to evade death. His return to the land of the living is covered here.
His girlfriend, Lily Hollister, would eventually join the Goblin roster. Harry played an integral role in her arrest, that story is covered here.
Eventually Lily would give birth to Harry's second son, Stanley. And Stanley's birth would cause Harry to leave NYC in order to live a normal life. That story is covered here.